A directed graph G has 5 vertices, numbered 1 through 5. The 5 x 5 matrix A is the adjacency matrix for G. The matrices A2 and A3 are given below. 0 0 1 0 0 0 0 0 1 0 A² = 0 0 0 0 1 A3= 10100 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 10100 0 1 0 1 0 0 0 1 0 1 Which edge is not in the transitive closure of G? (2,4) ○ (5,5) ○ (1,3) ○ (4,5)

Use the standard matrix for counterclockwise rotation in R2 to rotate the triangle with vertices (3,5), (5,3) and (3,0) counterclockwise 90 about the origin. Graph the triangles.

Let TA: 23 be the matrix transformation corresponding to A=[311124]. Find TA(u) and TA(v), where u=[12] v=[32].

Consider the matrices R=[ 0110 ] H=[ 1001 ] V=[ 1001 ] D=[ 0110 ] T=[ 0110 ] in GL(2,), and let G={ I2,R,R2,R3,H,D,V,T }. Given that G is a group of order 8 with respect to multiplication, write out a multiplication table for G. Sec. 3.3,22b,32b Find the center Z(G) for each of the following groups G. b. G={ I2,R,R2,R3,H,D,V,T } in Exercise 36 of section 3.1. Find the centralizer for each element a in each of the following groups. b. G={ I2,R,R2,R3,H,D,V,T } in Exercise 36 of section 3.1 Sec. 4.1,22 22. Find an isomorphism from the octic group D4 in Example 12 of this section to the group G={ I2,R,R2,R3,H,D,V,T } in Exercise 36 of Section 3.1. Sec. 4.6,14 14. Let G={ I2,R,R2,R3,H,D,V,T } be the multiplicative group of matrices in Exercise 36 of section 3.1, let G={ 1,1 } under multiplication, and define :GG by ([ abcd ])=adbc. Assume that is an epimorphism, and find the elements of K= ker . Write out the distinct elements of G/K. Let :G/KG be the isomorphism described in the proof of Theorem 4.27, and write out the values of .